METRIC PROPERTIES OF NON-RENORMALIZABLE S-UNIMODAL MAPS .1. INDUCED EXPANSION AND INVARIANT-MEASURES

For an arbitrary non-renormalizable unimodal map of the interval, f: I --> I, with negative Schwarzian derivative, we construct a related map F defined on a countable union of intervals DELTA. For each interval DELTA, F restricted to DELTA is a diffeomorphism which coincides with some iterate of f and whose range is fixed subinterval of I. If F satisfies conditions of the Folklore Theorem, we call f expansion inducing. Let c be a critical point of f. For f satisfying f'' (c) ] 0, we give sufficient conditions for expansion inducing. One of the consequences of expansion inducing is that Milnor's conjecture holds for f: the omega-limit set of Lebesgue almost every point is the interval [f2(c), F(c)]. An important step in the proof is a starting condition in the box case: if for initial boxes the ratio of their sizes is small enough, then subsequent ratios decrease at least exponentially fast and expansion inducing follows.